In this book, the term solar dynamo refers to the complex of mechanisms that cause the magnetic phenomena in the solar atmosphere. Usually, however, that complex is broken down into three components: (1) the generation of strong, large-scale fields of periodically reversing polarity, (2) the rise of these fields to the photosphere, and (3) the processing in, spreading across, and removal from the photosphere of magnetic flux. Components (2) and (3) are discussed in Chapters 4–6; in this chapter, we concentrate on aspect (1). Even on this limited topic, there is a stream of papers, but, as Rüdiger (1994) remarked, “it is much easier to find an excellent… review about the solar dynamo… than a working model of it.”

In dynamo theory, the mean, large-scale solar magnetic field is usually taken to be the axially symmetric component of the magnetic field that can be written, without loss of generality, as the sum of a toroidal (i.e., azimuthal) component Bφ ≡ (0, Bφ, 0) and a poloidal component, which is restricted to meridional planes: Bp ≡ (Br, 0, Bθ′), where θ′ is the colatitude. The poloidal component is usually pictured as if a dipole field aligned with the rotation axis were its major component, which is a severe restriction.

All solar-cycle dynamo models rely on the differential rotation v0(r, θ′) to pull out the magnetic field into the toroidal direction, as sketched in Fig. 6.10a; about this mechanism there is no controversy.

Introduction

On the main sequence, it has long been known that large mean rotational velocities are common among the early-type stars and that these velocities decline steeply in the F-star region, from 150 km s−1 to less than 10 km s−1 in the cooler stars (see Figure 1.6). As was shown in Section 6.3.2, the observed projected velocities indicate that the mean value of the total angular momentum 〈J〉 closely follows the simple power law 〈J〉 α M2 for stars earlier than spectral type F0, which corresponds to about 1.5M⊙ (see Figure 6.7). The difficulty is not to account for such a relation, which probably reflects the initial distribution of angular momentum, but to explain why it does not apply throughout the main sequence. It has been suggested that the break in the mean rotational velocities beginning at about spectral type F0 might be due to the systematic occurrence of planets around the low-mass stars (M ≲ 1.5M⊙), with most of the initial angular momentum being then transferred to the planets. Although this explanation has retained its attractiveness well into the 1960s, there is now ample evidence that it is not the most likely cause of the remarkable decline of rotation in the F-star region along the main sequence. Indeed, following Schatzman's (1962) original suggestion, there is now widespread agreement that this break in the rotation curve can be attributed to angular momentum loss through magnetized winds and/or sporadic mass ejections from stars with deep surface convection zones.

In this final chapter we present a synopsis of the observational constraints on dynamo processes in stars with convective envelopes that complements our review of studies of the solar dynamo in Chapter 7. We do not try to summarize the rapidly growing literature on mathematical and numerical models of stellar dynamos, but rather we attempt to capture the observational constraints on dynamos in a set of propositions, following Schrijver (1996). You will encounter some speculative links that attempt to bring together different facets of empirical knowledge, but we shall always distinguish conclusions from hypotheses.

Throughout this book, we use the term dynamo in a comprehensive sense, implying the ensemble of processes leading to the existence of a magnetic field in stellar photospheres, which evolves on times scales that are very short compared to any of the time scales for stellar evolution or for large-scale resistive dissipation of magnetic fields. Such a dynamo involves the conversion of kinetic energy in convective flows into magnetic energy.

Solar magnetic activity is epitomized by the existence of small-scale (compared to the stellar surface area), long-lived (compared to the time scale of the convective motions in the photosphere), highly structured magnetic fields in the photosphere, associated with nonthermally heated regions in the outer atmosphere, in which the temperatures significantly exceed that of the photosphere. Other cool stars exhibit similar phenomena, which are collectively referred to as stellar magnetic activity.

Outer-atmospheric imaging

The nearest cool star confronted us with the reality that cool stars have extremely inhomogeneous outer atmospheres. This was first confirmed for stars other than the Sun by the modulation of broadband signals, caused by starspots, and later by the discovery of the quasi-periodic variation in the Ca II H+K signal of some cool stars by Vaughan et al. (1981) caused by the rotation of an inhomogeneously covered stellar surface. Insight in stellar dynamos requires observational data on the properties of stellar active regions and their emergence patterns. For instance, we would like to know the sizes of stellar active regions and their lifetimes, the details of the structure of starspots, the emission scale height at different temperatures, and so on. In fact, we would like to know the entire three-dimensional geometrical structure of the outer atmospheres of cool stars. For that knowledge to be obtained, stellar surfaces should somehow be imaged by sounding the atmosphere from the photosphere on out. We would like to learn all this not merely for stars with activity levels similar to that of the Sun, but also for other stars, from the extremely active, tidally interacting binary systems for which much of the surface seems to be covered by areas as bright as solar active regions with a small fraction being even brighter, down to the very slowly rotating giant stars whose average coronal brightness is well below that of a solar coronal hole.

The Sun serves as the source of inspiration and the touchstone in the study of stellar magnetic activity. The terminology developed in observational solar physics is also used in stellar studies of magnetic activity. Consequently, this first chapter provides a brief illustrated glossary of nonmagnetic and magnetic features, as they are visible on the Sun in various parts of the electromagnetic spectrum. For more illustrations and detailed descriptions, we refer to Bruzek and Durrant (1977), Foukal (1990), Golub and Pasachoff (1997), and Zirin (1988).

The photosphere is the deepest layer in the solar atmosphere that is visible in “white light” and in continuum windows in the visible spectrum. Conspicuous features of the photosphere are the limb darkening (Fig. 1.1a) and the granulation (Fig. 2.12), a time-dependent pattern of bright granules surrounded by darker intergranular lanes. These nonmagnetic phenomena are discussed in Sections 2.3.1 and 2.5.

The magnetic structure that stands out in the photosphere comprises dark sunspots and bright faculae (Figs. 1.1a and 1.2b). A large sunspot consists of a particularly dark umbra, which is (maybe only partly) surrounded by a less dark penumbra. Small sunspots without a penumbral structure are called pores. Photospheric faculae are visible in white light as brighter specks close to the limb.

The chromosphere is the intricately structured layer on top of the photosphere; it is transparent in the optical continuum spectrum, but it is optically thick in strong spectral lines.

Introduction

As we may infer from the observations, most stars remain in a state of mechanical equilibrium, with the pressure-gradient force balancing their own gravitation corrected for the centrifugal force of axial rotation. Accordingly, theoretical work has tended to focus on the figures of equilibrium of a rotating star, assuming the motion to be wholly one of pure rotation. However, detailed study of the Sun has demonstrated the existence of large-scale motions in its convective envelope, both around the rotation axis and in meridian planes passing through the axis. Theoretical work has shown that largescale meridional currents also exist in the radiative regions of a rotating star. Moreover, as new results become available, it is becoming increasingly apparent that these regions contain a wide spectrum of turbulent motions embedded in the large-scale flow. All these problems are the domain of astrophysical fluid dynamics – a field that has developed quite slowly until recently.

Over the course of the past fifty years, however, meteorologists and oceanographers have made important advances in our knowledge of the behavior of rotating fluids. I thus find it appropriate to review some dynamical concepts that are applicable to both the Earth's atmosphere and the oceans. As we shall see, all of them play a key role in providing useful ideas for quantitative analysis of large-scale motions in a rotating star.

Although stellar rotation has aroused the interest of many distinguished astronomers and mathematicians for almost four hundred years, the theoretical study of the basic physical processes is largely a development of the twentieth century, indeed of the past thirty years or so. In this book I have attempted to present the theory of rotating stars as a branch of classical hydrodynamics, pointing out the differences and similarities between stars and other systems in which rotation is an essential ingredient, such as the Earth's atmosphere and the oceans. Throughout this volume I have thus assumed that the laws governing the internal dynamics of a rotating star are the usual principles of classical mechanics – basically mass conservation, Newton's second law of action, and the laws of thermodynamics. As is well known, one of the reasons why fluid motions in huge natural systems are so complex derives from the fact that the Navier–Stokes equation of motion is inherently nonlinear, so that the superposition of two solutions of a given problem is not necessarily a solution of that problem. In physical terms, this means that it is not possible in general to describe only the largest scale motions in a rotating star, since these flows will almost certainly interact with a whole spectrum of smaller-scale motions. The necessity of incorporating these small-scale, eddylike and/or wavelike motions into the large-scale flows remains as one of the important problems to be solved in astrophysical fluid dynamics.

The remainder of the main body of the book concerns observations and the ways in which they can be used to constrain the theoretical development we have made so far. The level of technical detail will be considerably less, because the observational situation doubtless will change. Roughly speaking, we shall be working from the largest scales to the smallest, beginning here with the cosmic microwave background (cmb).

Large angles and the COBE satellite

When we study cmb anisotropies on large angular scales, we are as close as one can get to directly studying the initial perturbations. In Section 2.4, we found that the Hubble length at the time of last scattering subtends an angle of around 1 deg on the last-scattering surface itself. On scales significantly larger than this, we are directly studying the effects of perturbations on scales greater than the Hubble length at the time of decoupling, which therefore have retained their primordial form.

The crucial observations in this regime are those of the Cosmic Background Explorer (COBE) satellite, taken over four years, which rightly can be said to have revolutionized cosmology. These observations provided, for the first time, an estimate of the spectrum of inhomogeneities in the Universe on very large scales, of order thousands of megaparsecs.

The COBE satellite carried three separate experiments. The Far Infra-Red Absolute Spectrometer (FIRAS; Mather et al. 1990) provided what is by far the most accurate measurement of the frequency spectrum of the microwave background, confirming it as a blackbody within experimental limits.